Question with divergence theorem

Question

Calculate flux of field $$\mathbf{F}=(3x^2 y+6)\mathbf{i}+\left(\frac{x y^2 +1}{3}\right)\mathbf{j}+(3yz^2+3)\mathbf{k}$$ in a box where it's opposite angles are in $(1,1,1)$ and $(2,2,2)$ and it's faces are axis orientated. My work: So it's a unit box and I guess divergence theorem should be used: $$\oint_S \mathbf{F}\bullet d\mathbf{S}=\iiint_V \nabla\bullet\mathbf{F} dV$$ So $ \nabla\bullet\mathbf{F} = \dfrac{20xy}{3}+6yz$ and $V=\{x,y,z \in [1,2]\}$ then $$\begin{align} \oint_S \mathbf{F}\bullet d\mathbf{S}&=\iiint_V \nabla\bullet\mathbf{F} dV \\ &= \int_1^2\int_1^2\int_1^2 \left(\dfrac{20xy}{3}+6yz\right) \; dxdydz \\ &=\frac{57}{2}\end{align}$$ Is my work correct?

Answer 1

I checked the differentiation, and the integral and everything seemed correct, the only thing that seemed shakey was the domain $1\le x,y,z\le2$, but if it's unit box this seems okay.

Best.